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In this paper, we investigate the stochastic evolution equations (SEEs)driven by $\log$-Whittle-Mat$\acute{{\mathrm{e}}}$rn (W-M) random diffusioncoefficient field and $Q$-Wiener multiplicative force noise. First, thewell-posedness of the underlying equations is established by proving theexistence, uniqueness, and stability of the mild solution. A sampling approachcalled approximation circulant embedding with padding is proposed to sample therandom coefficient field. Then a spatio-temporal discretization method based onsemi-implicit Euler-Maruyama scheme and finite element method is constructedand analyzed. An estimate for the strong convergence rate is derived. Numericalexperiments are finally reported to confirm the theoretical result.

An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise